TwoDirectionsSurface < WildSurfaces

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WildSurfaces.TwoDirectionsSurfacer1.2 - 07 Feb 2006 - 10:05 - ThierryMonteiltopic end

The two-directions surface

Description

2directions.mini.jpg

Identify the horizontal lines by vertical translation, and identify the vertical lines with the same style by translation. The lenght $\alpha$ is a positive irrational number.

id card

genus : 2
stratum : $\mathcal{H}(2)$
Veech group : at least a parabolic element in the vertical direction

Properties

This surface is completely periodic in the vertical direction as well as in the direction (2,-1). Using those two cylinder decompositions, one can compute its J-invariant and show that this surface has the SimpleJinvariantProperty

It is not completely periodic (see the horizontal direction); in particular it cannot be a torus branched covering.

One particularity of this surface is that it shows that in general, the cylinders in two completely periodic directions may not suffice to get all the homology (even if they suffice to get the J-invariant).
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I	Attachment	Action	Size	Date	Who	Comment
	2directions.mini.jpg	manage	2.2 K	06 Aug 2005 - 23:42	ThierryMonteil
	2directions.fig	manage	1.2 K	06 Aug 2005 - 23:45	ThierryMonteil

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